A collection of functions for finding the roots of polynomials in JavaScript.

roots.coffee

findRoots = (y = 0, coefficients = []) ->

    switch coefficients.length

        when 0

            return null

        when 1

            return uniFormula y, coefficients

        when 2

            return linierFormula y, coefficients

        when 3

            return quadraticFormula y, coefficients

        when 4

            return cubicFormula y, coefficients

        when 5

            return polyFormula y, coefficients

        else

            # Credit: https://math.stackexchange.com/a/200622/880755
            return "No general formula exists for polynomials of degree ≥ 5, see Abel–Ruffini theorem: https://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem"

uniFormula = (y = 0, coefficients = []) ->

    if y == coefficients[0]

        return Infinity

    else

        return null

linierFormula = (y = 0, coefficients = []) ->

    return [(y - coefficients[0]) / coefficients[1]]

# Credit: https://stackoverflow.com/a/33454565/1544937
quadraticFormula = (y = 0, coefficients = []) ->

    roots = []

    a = coefficients[2]
    b = coefficients[1]
    c = coefficients[0] - y

    positiveRoot = (-b + Math.sqrt(Math.pow(b, 2) - (4 * a * c))) / (2 * a)
    negativeRoot = (-b - Math.sqrt(Math.pow(b, 2) - (4 * a * c))) / (2 * a)

    if not isNaN positiveRoot then roots.push positiveRoot
    if not isNaN negativeRoot then if negativeRoot != positiveRoot then roots.push negativeRoot

    return roots

# Credit: https://stackoverflow.com/a/27176424/1544937
cubicFormula = (y = 0, coefficients = []) ->

    roots = null

    a = coefficients[3]
    b = coefficients[2]
    c = coefficients[1]
    d = coefficients[0] - y

    cuberoot = (x) ->

        y = Math.pow(Math.abs(x), 1/3)

        return if x < 0 then -y else y

    if Math.abs(a) < 1e-8 # Quadratic case, ax^2 + bx + c = 0

        a = b
        b = c
        c = d

        if Math.abs(a) < 1e-8 # Linear case, ax + b = 0

            a = b
            b = c

            if Math.abs(a) < 1e-8 # Degenerate case

                return []

            return [-b / a]

        D = (b * b) - (4 * a * c)

        if Math.abs(D) < 1e-8

            return [-b / (2 * a)]

        else if (D > 0)

            return [(-b + Math.sqrt(D)) / (2 * a), (-b - Math.sqrt(D)) / (2 * a)]

        return []

    p = (3*a*c - b*b) / (3*a*a)
    q = (2*b*b*b - 9*a*b*c + 27*a*a*d) / (27*a*a*a)

    if Math.abs(p) < 1e-8

        roots = [cuberoot(-q)]

    else if Math.abs(q) < 1e-8

        roots = [0].concat(if p < 0 then [Math.sqrt(-p), -Math.sqrt(-p)] else [])

    else

        D = q*q/4 + p*p*p/27

        if Math.abs(D) < 1e-8

            roots = [-1.5*q/p, 3*q/p]

        else if D > 0

            u = cuberoot(-q/2 - Math.sqrt(D))

            roots = [u - p / (3*u)]

        else

            u = 2 * Math.sqrt(-p / 3)
            t = Math.acos(3*q/p/u) / 3
            k = 2 * Math.PI / 3

            roots = [u * Math.cos(t), u * Math.cos(t-k), u * Math.cos(t-2*k)]

    for root in roots

        root -= b / (3*a)

    return roots

# Credit: https://stackoverflow.com/q/27217113/1544937
polyFormula = (y = 0, coefficients = []) ->

    return "https://en.wikipedia.org/wiki/Quartic_function"

roots.js

// Generated by CoffeeScript 2.7.0

var cubicFormula, findRoots, linierFormula, polyFormula, quadraticFormula, uniFormula;

findRoots = function(y = 0, coefficients = []) {
  switch (coefficients.length) {
    case 0:
      return null;
    case 1:
      return uniFormula(y, coefficients);
    case 2:
      return linierFormula(y, coefficients);
    case 3:
      return quadraticFormula(y, coefficients);
    case 4:
      return cubicFormula(y, coefficients);
    case 5:
      return polyFormula(y, coefficients);
    default:
      // Credit: https://math.stackexchange.com/a/200622/880755
      return "No general formula exists for polynomials of degree ≥ 5, see Abel–Ruffini theorem: https://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem";
  }
};

uniFormula = function(y = 0, coefficients = []) {
  if (y === coefficients[0]) {
    return 2e308;
  } else {
    return null;
  }
};

linierFormula = function(y = 0, coefficients = []) {
  return [(y - coefficients[0]) / coefficients[1]];
};

// Credit: https://stackoverflow.com/a/33454565/1544937
quadraticFormula = function(y = 0, coefficients = []) {
  var a, b, c, negativeRoot, positiveRoot, roots;
  roots = [];
  a = coefficients[2];
  b = coefficients[1];
  c = coefficients[0] - y;
  positiveRoot = (-b + Math.sqrt(Math.pow(b, 2) - (4 * a * c))) / (2 * a);
  negativeRoot = (-b - Math.sqrt(Math.pow(b, 2) - (4 * a * c))) / (2 * a);
  if (!isNaN(positiveRoot)) {
    roots.push(positiveRoot);
  }
  if (!isNaN(negativeRoot)) {
    if (negativeRoot !== positiveRoot) {
      roots.push(negativeRoot);
    }
  }
  return roots;
};

// Credit: https://stackoverflow.com/a/27176424/1544937
cubicFormula = function(y = 0, coefficients = []) {
  var D, a, b, c, cuberoot, d, i, k, len, p, q, root, roots, t, u;
  roots = null;
  a = coefficients[3];
  b = coefficients[2];
  c = coefficients[1];
  d = coefficients[0] - y;
  cuberoot = function(x) {
    y = Math.pow(Math.abs(x), 1 / 3);
    if (x < 0) {
      return -y;
    } else {
      return y;
    }
  };
  if (Math.abs(a) < 1e-8) { // Quadratic case, ax^2 + bx + c = 0
    a = b;
    b = c;
    c = d;
    if (Math.abs(a) < 1e-8) { // Linear case, ax + b = 0
      a = b;
      b = c;
      if (Math.abs(a) < 1e-8) { // Degenerate case
        return [];
      }
      return [-b / a];
    }
    D = (b * b) - (4 * a * c);
    if (Math.abs(D) < 1e-8) {
      return [-b / (2 * a)];
    } else if (D > 0) {
      return [(-b + Math.sqrt(D)) / (2 * a), (-b - Math.sqrt(D)) / (2 * a)];
    }
    return [];
  }
  p = (3 * a * c - b * b) / (3 * a * a);
  q = (2 * b * b * b - 9 * a * b * c + 27 * a * a * d) / (27 * a * a * a);
  if (Math.abs(p) < 1e-8) {
    roots = [cuberoot(-q)];
  } else if (Math.abs(q) < 1e-8) {
    roots = [0].concat(p < 0 ? [Math.sqrt(-p), -Math.sqrt(-p)] : []);
  } else {
    D = q * q / 4 + p * p * p / 27;
    if (Math.abs(D) < 1e-8) {
      roots = [-1.5 * q / p, 3 * q / p];
    } else if (D > 0) {
      u = cuberoot(-q / 2 - Math.sqrt(D));
      roots = [u - p / (3 * u)];
    } else {
      u = 2 * Math.sqrt(-p / 3);
      t = Math.acos(3 * q / p / u) / 3;
      k = 2 * Math.PI / 3;
      roots = [u * Math.cos(t), u * Math.cos(t - k), u * Math.cos(t - 2 * k)];
    }
  }
  for (i = 0, len = roots.length; i < len; i++) {
    root = roots[i];
    root -= b / (3 * a);
  }
  return roots;
};

// Credit: https://stackoverflow.com/q/27217113/1544937
polyFormula = function(y = 0, coefficients = []) {
  return "https://en.wikipedia.org/wiki/Quartic_function";
};